Civil Engineering Reference
In-Depth Information
To simplify Eq. (3.12), let
,
0 I
−
M
−1
K
−
M
−1
C
0
−
h
A
=
H
=
ð3
:
14Þ
where
A
is the 2
n
×2
n
state transition matrix in the continuous form, and
H
is the 2
n
× 3 ground
motion transition matrix in the continuous form. Then Eq. (3.12) becomes
z
ðÞ=
Az
ðÞ+
Ha
ðÞ
ð3
:
15Þ
Solving for the first-order linear differential equation in Eq. (3.15) gives
Þ
z
t
ðÞ+
e
A
t
ð
t
t
o
z
ðÞ=
e
A
t
−
t
o
ð
e
−
A
s
Ha
ðÞ
ds
ð3
:
16Þ
where
t
o
is the time of reference when the integration begins, which is typically the time when
the states
z
(
t
o
) are known, such as the initial conditions.
In order to integrate the term on the right-hand side of Eq. (3.16), the ground acceleration
vector
a
(
s
) should be expressed in a form of a continuous function. However, this ground
acceleration vector is typically digitized numerically in a short time interval for the duration
of the actual earthquake ground motion. Therefore, the integral needs to be evaluated numer-
ically. Let
t
k
+1
=
t
,
t
k
=
t
o
, and
Δ
t
=
t
k
+1
−
t
k
, and the subscript
k
denotes the
k
th time step, then it
follows from Eq. (3.16) that
z
k
+1
=
e
A
Δ
t
z
k
+
e
A
t
k
+1
ð
t
k
+1
t
k
e
−
A
s
Ha
ðÞ
ds
ð3
:
17Þ
Note that the objective of integration in Eq. (3.17) is to compute the area underneath the
e
−
A
s
Ha
(
s
) curve from
t
k
to
t
k
+1
, with a small time step. Because it is the area underneath
the curve that is of interest, any reasonable method of approximating the curve for
a
(
s
) between
the two time steps (e.g., constant acceleration or linear acceleration) will give a reasonable
approximation to the calculation of the enclosed area. In this presentation, the delta forcing
function approximation is used, where the ground acceleration vector takes the form:
a
ðÞ=
a
k
δ
s
−
t
k
ð
Þ
Δ
t
,
t
k
≤
s
<
t
k
+1
ð3
:
18Þ
Substituting Eq. (3.18) into Eq. (3.17) and performing the integration gives
z
k
+1
=
e
A
Δ
t
z
k
+
e
A
t
k
+1
ð
t
k
+1
t
k
e
−
A
s
Ha
k
δ
s
−
t
k
ð
Þ
Δ
tds
ð3
:
19Þ
=
e
A
Δ
t
z
k
+
e
A
t
k
+1
e
−
A
t
k
Ha
k
Δ
t
=
e
A
Δ
t
z
k
+
Δ
t
e
A
Δ
t
Ha
k
where
z
k
and
a
k
are the discretized forms of
z
(
t
) and
a
(
t
), respectively. Let
F
d
=
e
A
Δ
t
,
H
d
=
e
A
Δ
t
H
Δ
t
ð3
:
20Þ
